Approximate Representation of the p-norm Distribution

In surveying data processing,we generally suppose that the observational errors distribute normally.In this case the method of least squares can give the minimum variance unbiased estimation of the parameters.The method of least squares does not have the character of robustness.So the use of it will become unsuitable when a few measurements inheriting gross error mixed with others.We can use the robust estimate methods that can avoid the influence of gross errors.This kind of methods does not need to know the exact distribution of the observations.But it will also cause other difficult such as the hypothesis testing for estimated parameters when the sample size is not so big.For non-normally distributed measurements we can suppose they obey the p-norm distribution law. p-norm distribution is a distributive class which includes the most frequently used distributions such as the Laplace,normal and rectangular ones.This distribution is symmetry and has a kurtosis between 3 and -6/5 when p is greater than 1.Using p-norm distribution to describe the statistical character of the errors,the only assumption is that the error distribution is symmetry and has only one peak value.So the method of the p-norm distributive maximum likelihood adjustment can avoid determining the particular distributive model exactly before the data processing.It can fix the unknown parameters and the errors distribution simultaneously.This method possesses the property of a kind of self-adapting.But the density function of the p-norm distribution is so complex,which makes the theoretical analysis more difficult.And the troublesome calculation is also makes this method not suitable for practice.The research of this paper indicates that the p-norm distribution can be represented by the linear combination of Laplace distribution and normal distribution or by the linear combination of normal distribution and rectangular distribution approximately.Which kind of representation will be taken is according to that the parameter p is greater than 1 and lesser than 2 or that p is greater than 2.The approximate distribution have the same first four order moments with the exact one.That means approximate distribution have the same mathematical expectation,variance,skewness and kurtosis with p-norm distribution.Because of every density function used in the approximate formulas has a simple form,using the approximate density function to replace the p-norm ones will simplify the problems of p-norm distributed data processing obviously.